In higher category theory

A binary relation from a setXX to a set YY is called entire if every element of XX is related to at least one element of YY. This includes most examples of what the pre-Bourbaki literature calls a (total) multi-valued function (although that term usually implied some continuity or analyticity properties as well). An entire relation is sometimes called total, although that has another meaning in the theory of partial orders; see total relation.

Such a span is a relation iff the pairing map from the graphΓr\Gamma_r to X×YX \times Y is an injection, and such a relation is entire iff the projection map πr\pi_r is a surjection.

The axiom of choice says precisely that every entire relation contains a function. Failing that, the COSHEP axiom may be interpreted to say that, given XX, there is a single surjection πX:ΓX→X\pi_X: \Gamma_X \to X such that every entire relation from XX contains a relation given by a span whose left leg is πX\pi_X. In any case, entire relations may be preferable to functions in some contexts where the axiom of choice fails.